The radiative capture cross section of a highly pure (99.999%), 6.125(2) grams 238U sample ((9.56±05)e-4 atoms/barn) has been measured in the 185 m flight path at the CERN neutron time-of-flight facility n_TOF in the energy range 0.3 eV-20 keV. The Total Absorption Calorimeter (TAC) detection system, an array of 40 BaF2 crystals, was used utilising the total absorption technique, where all the γ-rays from the capture cascades are detected. These data have undergone careful background subtraction, with special care being given to the background originating from neutrons scattered by the 238U sample. Pile-up and dead-time effects have been corrected for using an innovative correction method valid for variable high count rates within a complex detection system such as the TAC. The resulting capture yield has an uncertainty of up to 2.2% below 5 keV and up to 3.7% between 5 and 20 keV, in line with the accuracy requested in the NEA High Priority Request List. A resonance analysis has been performed up to 5 keV with the code SAMMY. Between 5 and 20 keV, the cross section is treated as unresolved and averaged. The results confirm the excellent quality of the most recent nuclear data evaluations within quoted uncertainties and suggest some improvements. In particular, these TAC data suggest a 2% increase in the average cross section in the energy range 2-9 keV and a 5% reduction in the energy range 9-20 keV.